Given $a,b,c\in \Bbb{N}$ such that $\{a,b,c\}$ are coprime natural numbers and $a,b,c>1$. When
$$\frac{a^2}{b+c}＋\frac{b^2}{c+a}＋\frac{c^2}{a+b}\in\mathbb Z\,?$$
I know the solution $\{183,77,13\}$. Is there any other solution?

1 Answer
1

Yes, there is another solution. The next one I found is a bit big, namely
$$ a = 15349474555424019, b = 35633837601183731, c = 105699057106239769. $$

This solution also satisfies the property that
$$ \frac{a^{2}}{b+c} + \frac{b^{2}}{a+c} + \frac{c^{2}}{a+b} = \frac{31}{21} (a+b+c),
$$
which was true of $a = 13$, $b = 77$ and $c = 183$. If you clear denominators in the equation above, you find that both sides are multiples of $a+b+c$, and dividing out gives a plane cubic.

One wishes to search for rational points on this plane cubic, which by scaling can be assumed to be relatively prime integers. We also need points where $a, b, c > 0$ and $a+b+c$ is a multiple of $21$. This plane cubic is isomorphic to the elliptic curve
$$ E : y^2 + xy + y = x^3 - 8507979x + 9343104706, $$
and $E(\mathbb{Q})$ has rank $3$ and the torsion subgroup is $\mathbb{Z}/6\mathbb{Z}$. Let $E(\mathbb{Q}) = \langle T, P_{1}, P_{2}, P_{3} \rangle$ where $6T = 0$. I searched for points of the form $c_{1}T + c_{2}P_{1} + c_{3}P_{2} + c_{4}P_{3}$ where $0 \leq c_{1} \leq 5$ and $|c_{2}|, |c_{3}|, |c_{4}| \leq 5$. Of these $7986$ points, there are $1464$ where $a$, $b$ and $c$ are all positive, and these yield $11$ solutions (where $a < b < c$), the smallest of which is $(13,77,183)$. The second smallest is the one I gave above. It should be straightforward to prove that one can get infinitely many solutions in this way.

EDIT: I searched for solutions that satisfy
$$ \frac{a^{2}}{b+c} + \frac{b^{2}}{a+c} + \frac{c^{2}}{a+b} = t(a+b+c) $$
for all rational numbers $t = d/e$ with $d, e \leq 32$. This yields a number of smaller solutions. In particular, for $t = 9/5$ the rank of the curve is $2$ and one finds
$$ a = 248, b = 2755, c = 7227. $$
For $t = 21/17$ the rank is $2$ and one finds
$$ a = 35947, b = 196987, c = 401897. $$
Interestingly, $t = 31/21$ is the only case where the rank is $\geq 3$.

$\begingroup$Thank you for your answer. I understand how this problem is difficult and interesting !$\endgroup$
– MAEA2Mar 11 '17 at 6:22

$\begingroup$Jeremy, did you get the isomorphism by hand? Which tool did you use to find the rank of the curve?$\endgroup$
– guestMar 11 '17 at 7:49

$\begingroup$I used Magma to get the isomorphism and compute generators. Sage probably would do both parts work as well (especially since $E$ has a rational $2$-torsion point).$\endgroup$
– Jeremy RouseMar 11 '17 at 13:36